This research establishes critical process–property relationships for the Fused Deposition Modeling (FDM) of rigid (PLA) and functional flexible (TPU) polymers. By systematically evaluating the influence of key manufacturing parameters—infill density, print orientation, and loading rate—this study quantifies the structural integrity and operational limits of 3D-printed components. Tensile characterization, conducted according to ASTM D638-14 standards, reveals that horizontal and laid manufacturing strategies produce superior mechanical properties, whereas vertical configurations exhibit significant anisotropy due to interlayer bonding constraints. Stress-strain analysis differentiates the brittle, high-stiffness response of PLA from the hyper-elastic and viscoelastic behavior of TPU, highlighting the latter's sensitivity to rate-dependent processing. A critical finding for advanced manufacturing is the role of support-induced interfaces: while negligible for rigid PLA, support structures are shown to be detrimental to TPU, reducing maximum elongation by over 70% and necessitating support-free design strategies for functional elastomeric parts. Higher infill densities consistently enhance structural performance, providing a clear roadmap for optimizing material efficiency. These findings provide a principled framework for Design-for-Manufacturing (DfM) optimization, enabling the reliable production of tailored components for aerospace, biomedical, and soft-robotic applications.